> This collection forms an axiom system for the Natural numbers and all true statements are provable in this system. Indeed, every true statement is an axiom.
You have defined the axiom system as the set of all true statements about natural numbers so of course all true statements are provable! But crucially ...
> The problem with doing this is that the collection of all true statements about the Natural numbers is not recursively enumerable.
We want an extensional definition of a set of axioms. One way to get that is to enumerate it! This is important because ...
> Recursively enumerable was the goal of Hilbert and others because they wanted to reduce mathematics to an algorithmic or mechanical process of verification.
I can sympathize with this goal being that I use a proof assistant quite frequently.
Thank you for taking the time to respond!